While looking for something else, I came across a reference to a recent book called "The Topos of Music: Geometric Logic of Concepts, Theory and Performance" by G. Mazzola, published by Birkh"auser. Has anyone on the categories list seen this book? If so, can you say whether it is really about toposes and geometric logic as I (and most category-theorists) would understand those terms, or is it just a coincidence of terminology? Peter Johnstone
Well, I don't know, but the book is by Mazzola et al and one of the als is named Stefan Muller who, according to Amazon.ca has a paper published in The arithmetic and geometry of algebraic cycles: Proceedings of the CRM summer school, June 7-19, 1998, Banff, Alberta, Canada (CRM is the Centre de Recherche Mathematiques.) He has other publications that seem similarly mathematical, although there would seem to be a fringe aspect to some of them. One of his books, published this year, is called Complex Predicates. The other authors do not have other publicatons listed on Amazon. On Fri, 6 Dec 2002, Prof. Peter Johnstone wrote:
While looking for something else, I came across a reference to a recent book called "The Topos of Music: Geometric Logic of Concepts, Theory and Performance" by G. Mazzola, published by Birkh"auser. Has anyone on the categories list seen this book? If so, can you say whether it is really about toposes and geometric logic as I (and most category-theorists) would understand those terms, or is it just a coincidence of terminology?
Peter Johnstone
Prof. Peter Johnstone writes:
While looking for something else, I came across a reference to a recent book called "The Topos of Music: Geometric Logic of Concepts, Theory and Performance" by G. Mazzola, published by Birkh"auser. Has anyone on the categories list seen this book? If so, can you say whether it is really about toposes and geometric logic as I (and most category-theorists) would understand those terms, or is it just a coincidence of terminology?
Peter Johnstone
I've not yet seen the book, but I have looked at some of the related web sites. It is not "just a coincidence of terminology", though I don't understand if the usage of toposes, geometric logic, algebraic geometry, etc. is more than alegorical. -- Bob
On Fri, 6 Dec 2002, Prof. Peter Johnstone wrote:
While looking for something else, I came across a reference to a recent book called "The Topos of Music: Geometric Logic of Concepts, Theory and Performance" by G. Mazzola, published by Birkh"auser. Has anyone on the categories list seen this book? If so, can you say whether it is really about toposes and geometric logic as I (and most category-theorists) would understand those terms, or is it just a coincidence of terminology?
I have not seen the book, but a quick web search has been quite instructive[1,2,3,4,5]. The author is clearly writing about music theory, and deliberately using the language [2] and mathematics [3] of category and topos theory. The content of [3] seems to be more mathematics than music, but I doubt I can make much sense of either. It's hard for me to tell, but the evidence from the links below suggests that this is actually quite serious stuff, with an application of mathematics to the performance of music. References [1] Synopsis on amazon web site http://www.amazon.co.uk/exec/obidos/ASIN/3764357312/qid=1039250777/sr=1-1/re... [2] Mazzola, "Music Performance and Interpretation" http://www.engineeringandmusic.de/individu/mazzguer/maguproc.html [3] Mazzola, "Mathematical Music Theory -- Status Quo 2000" http://www.ircam.fr/equipes/repmus/mamux/documents/status.pdf [4] Homepage of "Engineering and Music" http://www.engineeringandmusic.de/ [5] Mazzola's own home page http://www.ifi.unizh.ch/staff/mazzola/
It just may, indeed, by our topoi and geometric logic: http://www.birkhauser.ch/books/math/5731.htm Mazzola, G., University of Zurich, Switzerland In Collaboration with Stefan Gvller and Stefan M|ller The Topos of Music Geometric Logic of Concepts, Theory, and Performance 2002. 1368 pages. Hardcover. incl. CD-ROM $ 98.- / CHF 144.- ISBN 3-7643-5731-2 Subscription price $ 98.- / CHF 144.- valid until 31.12.02 Regular Price: $ 128.- / CHF 188.- Since the Greek antiquity it has been a tradition of European thinking to describe musical facts in a mathematical language. This formal apparatus has always mirrored the status quo of mathematical knowledge and the requirements of current sound technology. The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Tvne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Tvne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples. Contributors: Carlos Agon, Moreno Andreatta, Girard Assayag, Jan Beran, Chantal Buteau, Roberto Ferretti, Anja Fleischer, Harald Fripertinger, Jvrg Garbers, Werner Hemmert, Michael Leyton, Emilio Lluis Puebla, Mariana Montiel Hernandez, Thomas Noll, Joachim Stange-Elbe, Hans Straub, Oliver Zahorka
On Fri, 6 Dec 2002, Prof. Peter Johnstone wrote:
While looking for something else, I came across a reference to a recent book called "The Topos of Music: Geometric Logic of Concepts, Theory and Performance" by G. Mazzola, published by Birkh"auser. Has anyone on the categories list seen this book? If so, can you say whether it is really about toposes and geometric logic as I (and most category-theorists) would understand those terms, or is it just a coincidence of terminology?
Peter Johnstone
To answer my own question, I have now managed to get hold of a copy of the book. It's a very large book (1360 pages, U.K. price 83.5 pounds) -- but I suppose it's not for me to complain about that. The author is indeed attempting to apply topos theory (and quite a lot of other high-powered mathematics) to the analysis of music. Whether there's any substance in it is hard to tell from a qick skim-through: I have a strong suspicion that the whole thing is skilfully crafted from bovine excrement, but I'll need to consult my colleagues in the Music Faculty to see whether they think there's anything in it. I am not cited in the Bibliography (naturally I checked this first!) but the books by Goldblatt and Mac Lane--Moerdijk are both there. Bill Lawvere is cited for ETCS (1964), but not for anything more recent. Alexander Grothendieck is cited for SGA4, and also for a private communication to the author, commenting on the book's German- language predecessor (published 1990): "Das ist wohl schon die Mathematik des 'Neuen Zeitalters'". Composers cited in the Bibliography are, in alphabetical order, Milton Babbitt, J.S. Bach, Pierre Boulez, Claude Debussy, J.J. Fux, Paul Hindemith, Sigfrid Karg-Elert, W.A. Mozart, Arnold Schoenberg, Franz Schubert, Robert Schumann and Anton Webern (I may have missed one or two). Other citations include Aristotle, Francis Bacon, Umberto Eco and (inevitably?) Douglas Hofstadter. Peter Johnstone
I obtained that thick book of Mazzola et al a few days ago. They try to analyse and synthesize various aspects of music with mathematical structures. They construct such structures from the category of modules and borrow some techniques from algebraic geometry, but formal logic does not appear in it. They also utilize works in computer science for their purpose and indeed has developed their software for musical composition and performance. It might be interesting as an application of mathematical conceptions. I would like to suspend my judgement on the value of this book until using the software. Hiroyuki Miyoshi
participants (6)
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Robert L. Knighten